On regular operators on Banach lattices


Banach lattice
regular operators
M-weakly compact operators
order continuous norm

How to Cite

Gok, O. (2023). On regular operators on Banach lattices. Acta Et Commentationes Exact and Natural Sciences, 14(2), 53-56. https://doi.org/10.36120/2587-3644.v14i2.53-56


Let $E$ and $F$ be Banach lattices and $X$ and $Y$ be Banach spaces. A linear operator $T: E \rightarrow F$ is called regular if it is the difference of two positive operators. $L_{r}(E,F)$ denotes the vector space of all regular operators from $E$ into $F$. A continuous linear operator $T: E \rightarrow X$ is called $M$-weakly compact operator if for every disjoint bounded sequence $(x_{n})$ in $E$, we have $lim_{n \rightarrow\infty} \| Tx_{n} \| =0$. $W^{r}_{M}(E,F)$ denotes the regular $M$-weakly compact operators from $E$ into $F$. This paper is devoted to the study of regular operators and $M$-weakly compact operators on Banach lattices. We show that $F$ has a b-property if and only if $L_{r}(E,F)$ has b-property. Also, $W^{r}_{M}(E,F)$ is a $KB$-space if and only if $F$ is a $KB$-space.
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